The flexibility of network modeling and the rapid growth of network data in recent years make it urgent to design effective network comparison methods for applications such as comparing diffusion cascade of news [1]}, the classification of proteins [2]} and evaluation of generative network models [3]}, [4]}, [5]}.
Researchers have proposed methods based on graph isomorphism to compare networks [6]}, [7]}, [8]}, [9]}. The main limitations of isomorphism based methods are as follows: first of all, isomorphism based methods can only compare networks with the same size and are not scalable to large networks with millions of nodes. Secondly, this kind of methods can only tell whether two networks are isomorphic or not but to what extent two networks are different
is hardly measured. Thanks to the mature research of network topology mining [10]}, [11]}, [12]}, [13]}, an amount of researchers have studied how to use network characteristics, e.g., adjacency matrix, node degree, shortest path distance, to compare networks with huge and different sizes. For instance, Saxena et al. [14]} introduced a network similarity method based on hierarchical diagram decomposition via using canberra distance, which considers both local and global network topology. Lu et al. [15]} proposed a manifold diffusion method based on random walk, which can not only distinguish networks with different degree distributions but also can discriminate networks with the same degree distribution. Beyond the direct comparison of network topology, we have witnessed the effectiveness of using quantum information science, i.e., information entropy, in network comparison.
For example, De Domenico et al. [16]} proposed a set of information theory tools for network comparison based on spectral entropy. Schieber et al. [17]} quantified the dissimilarities between networks by considering the probability distribution of the shortest path distance between nodes. Chen et al. [18]} proposed a comparison method based on node communicability and spectral entropy. The basic idea behind this kind of method is that one specific network property, such as the shortest path distance [17]} and node communicability matrix [18]} , is chosen to measure the information content of a network via a proper entropy. Therefore, the dissimilarity between two networks is given by the difference between the information content of them. However, we claim that the selection of one specific property as a representative of network information content may not be able to capture the information of a whole network.
